# Representation with a highest weight vector

A linear representation (cf. Representation of a Lie algebra) $\rho$ of a finite-dimensional semi-simple split Lie algebra $\mathfrak g$ over a field $k$ of characteristic zero with a split Cartan subalgebra $\mathfrak t$, having the following properties.

1) In the space $V$ of $\rho$ there is a cyclic vector $v$( i.e. $V$ is the smallest $\mathfrak g$- invariant subspace containing $v$).

2) $\rho ( h) v = \lambda ( h) v$ for all $h \in \mathfrak t$, where $\lambda$ is some fixed linear form on $\mathfrak t$ with values in $k$.

3) If $\alpha _ {1} \dots \alpha _ {r}$ is a system of simple roots, defined by a lexicographical order on the set $\Delta$ of all roots of $\mathfrak g$ relative to $\mathfrak t$( cf. Root system), and if $e _ {\alpha _ {i} } , \mathfrak t _ {\alpha _ {i} } , h _ {\alpha _ {i} }$ are the vectors from the Chevalley basis of $\mathfrak g$ corresponding to $\alpha _ {i}$, $i = 1 \dots r$, then $\rho ( e _ {\alpha _ {i} } ) ( v) = 0$ for all $i = 1 \dots r$. Thus, $\lambda$ is a weight relative to the restriction of $\rho$ to $\mathfrak t$( cf. Weight of a representation of a Lie algebra); it is called a highest weight. The space $V$ is called a cyclic $\mathfrak g$- module with highest weight $\lambda$ and generator $v$, and $v$ is called a highest weight vector.

There exists for every linear form $\lambda$ on $\mathfrak t$ a unique, up to equivalence, irreducible representation $\rho _ \lambda$ of $\mathfrak g$ with highest weight $\lambda$. The $\mathfrak g$- module $V ( \lambda )$ determined by $\rho _ \lambda$ is a direct sum of weight subspaces relative to the restriction of $\rho _ \lambda$ to $\mathfrak t$. Their weights have the form

$$\lambda - \sum _ {i = 1 } ^ { r } n _ {i} \alpha _ {i} ,$$

where the $n _ {i}$ are non-negative integers. The weight subspace $V _ \mu ( \lambda )$ of weight $\mu$ is finite-dimensional, spanned over $k$ by vectors of the form

$$( \rho _ \lambda ( f _ {\alpha _ {i _ {1} } } ) \dots \rho _ \lambda ( f _ {\alpha _ {i _ {s} } } ) ) ( v ) ,$$

and for any $h \in \mathfrak t$ the restriction of $\rho _ \lambda ( h)$ to $V _ \mu ( \lambda )$ is the operator of scalar multiplication by $\mu ( h)$. The space $V _ \lambda ( \lambda )$ is one-dimensional; the weight $\lambda$ is the only highest weight of $\rho _ \lambda$ and can be characterized as the unique weight of the $\mathfrak t$- module $V ( \lambda )$ such that any other weight has the form

$$\lambda - \sum _ {i = 1 } ^ { r } n _ {i} \alpha _ {i} ,$$

where the $n _ {i}$ are non-negative integers.

A representation $\rho _ \lambda$ is finite-dimensional if and only if $\lambda$ is a dominant linear form on $\mathfrak t$, i.e. $\lambda ( h _ {\alpha _ {i} } )$ is a non-negative integer for $i = 1 \dots r$. Every irreducible finite-dimensional linear representation of $\mathfrak g$ has the form $\rho _ \lambda$ for some dominant linear form $\lambda$ on $\mathfrak t$( hence all such representations are classified, up to equivalence, by the dominant linear forms on $\mathfrak t$). The set of all weights of a finite-dimensional representation $\rho _ \lambda$ relative to $\mathfrak t$ is invariant relative to the Weyl group of $\mathfrak g$( regarded as a group of linear transformations of $\mathfrak t$), and if weights $\mu$ and $\gamma$ belong to one orbit of the Weyl group, then the dimensions of the spaces $V _ \mu ( \lambda )$ and $V _ \gamma ( \lambda )$ are equal. For every weight $\mu$ and every root $\alpha \in \Delta$ the number $\mu ( h _ \alpha )$ is an integer; if, moreover, $\mu + \alpha$ is also a weight, then

$$\rho ( e _ \alpha ) ( V _ \mu ( \lambda )) \neq 0$$

(here $h _ \alpha$ is the element in $\mathfrak t$ corresponding to $\alpha$ and $e _ \alpha$ is the root vector of $\alpha$).

#### References

 [1] N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979)) [2] , Theórie des algèbres de Lie. Topologie des groupes de Lie , Sem. S. Lie , Ie année 1954–1955 , Secr. Math. Univ. Paris (1955) [3] D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian) [4] E. Cartan, "Les tenseurs irréductibles et les groupes linéaires simples et semi-simples" Bull. Sci. Math. , 49 (1925) pp. 130–152 [5] Harish-Chandra, "On some applications of the universal enveloping algebra of a semisimple Lie algebra" Trans. Amer. Math. Soc. , 70 (1951) pp. 28–96
How to Cite This Entry:
Representation with a highest weight vector. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Representation_with_a_highest_weight_vector&oldid=48523
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article